As I write this review in the spring of 2008, election season is marching onward, as is the list of election-related books that publishers are releasing in an attempt to cash in on the excitement. Mathematicians are certainly not immune from this phenomenon, and a number of new books on the mathematics of elections are being published. In fact, one of the leaders in this area of study, Steven J. Brams, has at least three books scheduled to appear in 2008. While I gave a mixed review to the re-release of his The Presidential Election Game, I am pleased to say that his latest book, Mathematics and Democracy: Designing Better Voting And Fair-Division Procedures , was much more to my liking.

Mathematics and Democracy is a collection of papers that the author wrote with various co-authors in the field of decision theory. While this means the book lacks a coherent narrative that might have been useful, it also means that each chapter is essentially self-contained and a reader can pick and choose which topics they wish to read about. The first seven chapters deal with voting procedures: Brams makes no attempt to hide his enthusiasm for approval voting, a system in which voters designate each candidate as acceptable or unacceptable and the candidate who is acceptable to the largest numbers of voters wins. The book has chapters dedicated to the theory underlying approval voting as well as case studies in which approval voting was used and the lessons which can be gleaned from those examples. There is a chapter devoted to a variation of approval voting which takes into account the preferences a voter may have among candidates they find acceptable, and several chapters devoted to situations where one might want to elect multiple winners from a single election.

The second half of the book is dedicated to questions of how to divide goods between people in a way which is fair. Of course, fairness is an ill-defined concept when different people place different values on a single good, and Brams considers several different goals which one could strive for in an attempt to make all of the parties as happy as possible. There are many variations on this question, depending on if we have one homogenous good which can be divided in any way one might want (such as a stack of money), a heterogeneous good such as a piece of cake where some people prefer pieces with lots of frosting and others prefer more chocolate, a number of goods of different values which are indivisible, or many other variants. Brams dedicates chapters to a number of these variants, and attempts to come up with systems to divide the goods which are fair.

The articles in Mathematics and Democracy all assume very little background beyond basic game theoretic concepts such as Nash Equilibria, yet they manage to have significant mathematical substance. Along with the self-contained nature of each chapter, this makes the book a fertile source for readings to learn about the field — or to give to interested students and colleagues. Brams gives many examples, some of which come from real life situations ranging from the 1978 Camp David negotiations between Egypt and Israel to the 1987 election of MAA officers, and others of which are cleverly constructed to make various theoretical points. His writing style is engaging and involved exactly the level of technical details which this reviewer wanted to read. The book also contains a thorough bibliography which can lead an interested reader to discover much more about the field of social choice theory, something which I imagine many readers of Brams' book will find themselves wanting to do.

Darren Glass is assistant professor of mathematics at Gettysburg College